(2) Let \(R=\mathbb {Z}\)  and consider the \(R\) -module \(N=\mathbb {Q}/\mathbb {Z}_{(2\mathbb {Z})}\) . By [1]},
\(A=R\propto N\)  is a Torch ring. Thus, every cyclic \(A\) -module is weakly IN by Theorem REF .
On the other hand, since \(R\)  is not a clean ring, \(A\)  is not clean by [2]}.
Therefore the ring \(A\)  has a cyclic \(A\) -module which is not strongly CS by Theorem REF .

